We introduce a topometric version of Lipschitz-free spaces and study its universal property. Another aim of this paper is to investigate actions of topological groups G on Lipschitz-free spaces <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2967_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {F}}(M)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, induced by isometric actions on pointed metric spaces M. In particular, we study the associated dynamical G-systems under the weak-star topology, focusing on the dual action on <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2967_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="136" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{Lip}_0(M) = {\mathcal {F}}(M)^*$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>Lip</mtext> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi mathvariant="script">F</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> and the bidual <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2967_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {F}}(M)^{**}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> <mrow /> <mrow> <mrow /> <mo>∗</mo> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </mrow> </math></EquationSource> </InlineEquation>.

Lipschitz-Free Spaces: A Topometric Approach and Group Actions

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2025年10月15日

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Michael Megrelishvili

摘要

We introduce a topometric version of Lipschitz-free spaces and study its universal property. Another aim of this paper is to investigate actions of topological groups G on Lipschitz-free spaces ${\mathcal {F}}(M)$ $F (M)$ , induced by isometric actions on pointed metric spaces M. In particular, we study the associated dynamical G-systems under the weak-star topology, focusing on the dual action on $\textrm{Lip}_0(M) = {\mathcal {F}}(M)^*$ ${Lip}_{0} (M) = F {(M)}^{*}$ and the bidual ${\mathcal {F}}(M)^{**}$ $F {(M)}^{* *}$ .